[/quote]
,
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
be an acute scalene triangle inscribed in a circle 
.
intersects 
at 
and 
.
is the midpoint of 
.
be the altitude in 
,
intersects 
.
be the midpoint of 
be the reflection of 
with respect to 
and 
are similar.
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.
with sum 
are such that the equation 
has a positive root 
smaller than 
.
such that the inequality 
holds.
and circumscribed about the incircle 
,
.
,
,
,
,
,
,
,
,
(distinct from 
intersects 
(distinct from 
of 
and 
at 
and 
,
at 
.
.
are the midpoints of the arcs 
and 
(not containing 
and 
,
passes through a fixed point, independent of the choice of 
,
,
,
and 
such that 
is a parallelogram and 
is a cyclic quadrilateral. Let 
be a line passing through 
at 
and intersects line 
at 
.
.
.
is divisibly by 
.
and 
is entered, the new password is 
,
be a circle and 
a number from a set 
is written --- in the first row numbers 
,
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
.![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](http://latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
lying outside the circle 
,
.
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
has circumcircle 
to 
Y by
Given 
with 
, let 
be the bisector of 
with 
on the side 
. Let 
be a circle passing through 
and which is tangent to at . Suppose cuts the side 
again at 
. The tangent to the circumcircle of 
at intersects again at 
. Let 
be the intersection point of the segments 
and 
. Prove that 
is perpendicular to .
with 
, let 
be the bisector of 
with 
on the side 
. Let 
be a circle passing through 
and which is tangent to at . Suppose cuts the side 
again at 
. The tangent to the circumcircle of 
at intersects again at 
. Let 
be the intersection point of the segments 
and 
. Prove that 
is perpendicular to .
.
are collinear.![\[
\measuredangle FDB = \measuredangle DEB = \measuredangle DEA = \measuredangle CDA
\]](http://latex.artofproblemsolving.com/8/7/4/8748bfe341d5a6a39e8d7f55d5a83345c9a96c0e.png)
so 
are reflections of each other over 
.
,
and 
so 
.
are reflections of each other over 
are collinear on ![[asy]
import graph; size(8cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -5.754031141961579, xmax = 15.23468239541721, ymin = -1.175589639470279, ymax = 10.235140387729096; /* image dimensions */
pen qqwuqq = rgb(0,0.39215686274509803,0);
/* draw figures */
draw(circle((5,5), 5), linewidth(0.4));
draw((0.3543809158514515,3.151156219418525)--(9.64561908414855,3.151156219418525), linewidth(0.4));
draw((3.4419996538801927,9.751066714064386)--(0.3543809158514515,3.151156219418525), linewidth(0.4));
draw((3.4419996538801927,9.751066714064386)--(9.64561908414855,3.151156219418525), linewidth(0.4));
draw((3.4419996538801927,9.751066714064386)--(5,0), linewidth(0.4));
draw(circle((4.496516368465568,6.53535545021987), 3.3841992308013458), linewidth(0.4) + qqwuqq);
draw(circle((2.4254486421585106,3.4820660910297927), 2.0973370902362065), linewidth(0.4) + blue);
draw((1.3521762622312536,5.2839843101967325)--(5.551033083050948,9.751066714064384), linewidth(0.4));
draw((4.49651636846557,8.629186816433327)--(4.496516368465569,3.1511562194185245), linewidth(0.4));
draw((3.4419996538801927,9.751066714064386)--(5.551033083050948,9.751066714064384), linewidth(0.4));
draw((1.3521762622312536,5.2839843101967325)--(4.496516368465569,3.1511562194185245), linewidth(0.4));
draw((1.3521762622312536,5.2839843101967325)--(7.640856474699886,5.283984310196731), linewidth(0.4));
draw((7.640856474699886,5.283984310196731)--(5.551033083050948,9.751066714064384), linewidth(0.4));
draw((4.496516368465569,3.1511562194185245)--(5.551033083050948,9.751066714064384), linewidth(0.4));
/* dots and labels */
dot((5,5),linewidth(3pt) + dotstyle);
label("$O$", (5.054152085121636,5.075833411734659), NE * labelscalefactor);
dot((3.4419996538801927,9.751066714064386),linewidth(3pt) + dotstyle);
label("$A$", (3.4975728514882816,9.820889462649252), NE * labelscalefactor);
dot((0.3543809158514515,3.151156219418525),linewidth(3pt) + dotstyle);
label("$B$", (0.40952050089307696,3.2305338363789846), NE * labelscalefactor);
dot((9.64561908414855,3.151156219418525),linewidth(3pt) + dotstyle);
label("$C$", (9.698783669350197,3.2305338363789846), NE * labelscalefactor);
dot((5,0),linewidth(3pt) + dotstyle);
label("$M$", (5.054152085121636,0.07971619410500982), NE * labelscalefactor);
dot((4.496516368465569,3.1511562194185245),linewidth(3pt) + dotstyle);
label("$D$", (4.552029751691522,3.2305338363789846), NE * labelscalefactor);
dot((4.496516368465568,6.53535545021987),linewidth(3pt) + dotstyle);
label("$O_{1}$", (4.552029751691522,6.607306528696512), NE * labelscalefactor);
dot((1.3521762622312536,5.2839843101967325),linewidth(3pt) + dotstyle);
label("$E$", (1.4012121094175531,5.364553753456976), NE * labelscalefactor);
dot((5.551033083050948,9.751066714064384),linewidth(3pt) + dotstyle);
label("$F$", (5.6064866518947625,9.820889462649252), NE * labelscalefactor);
dot((4.49651636846557,8.629186816433327),linewidth(3pt) + dotstyle);
label("$P$", (4.552029751691522,8.703667270767244), NE * labelscalefactor);
dot((7.640856474699886,5.283984310196731),linewidth(3pt) + dotstyle);
label("$G$", (7.690294335629739,5.364553753456976), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/1/d/c/1dcc15ffe783f92606d016cf86c69b0a64ece40f.png)
as the midpoint of arc 
,
and 
as the center of 
.
,
as the circle tangent to 
and tangent to the circumcircle at 
.
,
.
,
.
![\[\angle AFD = \angle BED = 180 - \angle EBD - \angle BDE = 180 - \angle FDE - \angle BDE = \angle FDC\]](http://latex.artofproblemsolving.com/f/3/8/f38b76fa9759c5912e1352144ba4128af3989114.png)
Thus we are done. 
,
is an isosceles trapezoid (by claim 
and 
)
and due to the fact that
however 
by the definition of 
.
